Worksheet 5

1. For what $\mathbf{b}$ does $A \mathbf{x} = \mathbf{b}$ have a solution if \(A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 2 \\ -1 & -3 & 1 \\ 1 & 0 & 1 \end{pmatrix} \ ?\)

2. Let \(A = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 2 \\ 1 & 0 & 2 \end{pmatrix}\)

   Row reduce $A$ and find a parametric presentation for $\mathcal Z(A)$. Use you parametric presentation of $\mathcal Z(A)$ to find a $B$ with \(\mathcal Z(A) = \mathcal R(B)\)

   Note this is the converse statement to what we saw in the notes which said that the range of any matrix is the null space of another matrix.

3. For any $A$ does there exist a $B$ such that \(\mathcal Z(A) = \mathcal R(B) \ ?\)
   Provide a proof or counterexample. This would say the null space of any matrix is equal to the range of another matrix.